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```Cycloidal Curves.

Pitch Circle, Li|ie, or Surface of a spur wheel
represents the contour of the ideal disc or straight-edge iuh£c2* *"^jt
transmit the same motion.

To transmit perfectly uniform motion the teeth rtiu& £
specially formed, and all teeth in gear at once must contr'i
to the perfection of the motion. To fulfil these condltioti
normals to all surfaces of contact must pass through the m^
point of the pitch lines (Fig. 492), and this is obtained whet>
tooth b c, on A, is the envelope of the relative positions of the o *t*
tooth on B (Fig. 493) when the discs are rolled together,
teeth are actually drawn, however, in a somewhat
manner. (See Appendix ///., p. 926.)

Cycloidal Curves.—A cycloid may be traced by   a
on the rim of a disc which rolls along a straight edge,
epi-cycloid when the disc  rolls  upon a circular arc  (Fig.
A  hypo-cycloid is   similarly  traced  within an  annular   d.ls^
at   Fig.  495,,  noting that  when  the rolling  disc   is     half     tJ
diameter of the annulus a straight line is obtained,    as
dotted;   a   fact  which  has  produced  White's   parallel
(Fig. 496).

Rolling Circle.—The above curves will serve for
teeth, if the same rolling circle be adopted for parts that
in contact, the tooth point being formed by epi-cycloids
the root by hypo-cycloids. Taking the wheels A and B, Fig.
a rolling circle is first to be chosen as governed by the
curves: thus, if the circle be half the pitch diameter, radial
are formed, as at c; if larger, the root will be undercut as at: \
and E is drawn with a circle of J-pitch diameter. The la 11
is reasonable, as giving strength, while yet avoiding ot>llc|
pressure on bearings. 'Adopting then the rolling circles sto^m
F may roll the root of B and the point of A, because these arc*
engage, but G will serve for root of A and point of B. Wlfcoxi ,
the wheels of a train are to work together interchangeably, t
same rolling circle must be used throughout If the toe
pressure is always in one9direction, as in Fig. 499, a large roll!
circle may be adopted for the acting surfaces and a small o:n.€i j
the back surfaces, thus giving great root strength without o"t>lic|
action.```